Numerical studies of recovery chances for a simplified EIT problem TU Chemnitz - Faculty of Mathematics Numerical studies of recovery chances for a simplified EIT problem Christopher Hofmann joint work with Bernd Hofmann and Roman Unger TU Chemnitz - Faculty of Mathematics 2nd November 2017 TUC · 2nd November 2017 · Christopher Hofmann 1 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Content 1. Motivation 2. General EIT Model 3. Simplified Model 4. Numerical Examples 5. Conclusions 6. References TUC · 2nd November 2017 · Christopher Hofmann 2 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Motivation Introduction to EIT What is EIT? ◮ Apply current on the specimen’s boundary. ◮ Take voltage measurements on the boundary of the specimen. ◮ Reconstruct electric conductivity within the specimen from these measurements. ◮ Practical applications mostly in medical imaging. ◮ Focus often lies on the detection of inclusions. Figure: Specimen to take EIT measurements. 1 1 Loyola, et al: Detection of spatially distributed damage in fiber-reinforced polymer composites TUC · 2nd November 2017 · Christopher Hofmann 3 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Motivation Detection of mechanical strain in CNT/CNF Case study motivated by an application on the detection of mechanical strain within a CNT/CNF specimen. Opposed to many previous approaches it is therefore assumed: ◮ ’Stripe’ structure with piecewise constant conductivities ◮ No a priori information on background conductivity TUC · 2nd November 2017 · Christopher Hofmann 4 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Motivation Overview of specimen Figure: Specimen with electrodes and finite element grid. TUC · 2nd November 2017 · Christopher Hofmann 5 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
General EIT Model General EIT Model ◮ Ω ∈ R 2 , with smooth boundary ∂ Ω ◮ Conductivity σ ( x ) , x ∈ Ω , electric potential u ( x ) , x ∈ Ω , ◮ σ∂ ν u | ∂ Ω can be interpreted as current ◮ Laplace equation holds in the interior of Ω : (+) ∇ · ( σ ( x ) ∇ u ( x )) = 0 TUC · 2nd November 2017 · Christopher Hofmann 6 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
General EIT Model Neumann-to-Dirichlet map For practical applications it is desirable to apply current and to measure voltages/potential differences: Λ σ : L 2 ⋄ ( ∂ Ω) → L 2 g | ∂ Ω �→ u g | ∂ Ω ⋄ ( ∂ Ω) , with u g weak solution of the Laplace equation with Neumann boundary values σ∂ ν u | ∂ Ω = g | ∂ Ω with L ∞ + (Ω) := { σ ∈ L ∞ (Ω) : inf x ∈ Ω σ ( x ) > 0 } and � L 2 ⋄ ( ∂ Ω) := { g ∈ L 2 ( ∂ Ω) : gds = 0 } . ∂ Ω TUC · 2nd November 2017 · Christopher Hofmann 7 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
General EIT Model Forward Operator The forward operator in this model situation is then given by A : L ∞ + (Ω) → L ( L 2 ⋄ ( ∂ Ω)) , σ �→ Λ σ . Inverse Problem: Retrieve σ ( x ) , x ∈ Ω , from data of the current-to-voltage map Λ σ . TUC · 2nd November 2017 · Christopher Hofmann 8 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Simplified Model Simplified and Discretized Model In practice it is obviously impossible to retrieve measurements on the whole boundary ∂ Ω of Ω . Therefore we assume: ◮ K electrodes ǫ k , � K k =1 ǫ k ⊂ ∂ Ω ◮ I k and U k : current and voltage on the k-th electrode ◮ Steady state: � K k =1 I k = 0 (in- and outgoing currents add up to zero) We further assume, that the conductivity σ is isotropic. TUC · 2nd November 2017 · Christopher Hofmann 9 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Simplified Model Electrode Modelling Shunt Electrode Model: ◮ No current flows outside electrodes: σ∂ ν u | ∂ Ω \ � K k =1 ǫ k = 0 ◮ Current on electrode ǫ k is equally distributed with overall current � I k = ǫ k σ∂ ν u | ∂ Ω ds . I k ◮ σ∂ ν u | ǫ k = | ǫ k | , with arclength | ǫ k | of the electrode ǫ k . ◮ It is further assumed that the potential on every electrode is constant, i.e. u | ǫ k = const . TUC · 2nd November 2017 · Christopher Hofmann 10 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Simplified Model Discretized Forward Operator The solution of Laplace problem (+) is not unique, it is assumed that the potentials add ⋄ = { x ∈ R K : � K up to zero as well. With R K k =1 x k = 0 } the mapping ( I k ) K k =1 ∈ R K ( U k ) K k =1 ∈ R K R σ : �→ ⋄ ⋄ is then the basis for required sets of measurements. TUC · 2nd November 2017 · Christopher Hofmann 11 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Simplified Model Discretized Forward Operator ◮ Discretize Ω ∈ R 2 with triangular mesh with 32 boundary edges. ◮ K = 8 electrodes ǫ i ( i = 1 , .., 8) ◮ Neumann boundary conditions on two neighbouring electrodes: σ∂ ν u | ǫ i = 1 and σ∂ ν u | ǫ i +1 = − 1 ◮ Set Dirichlet boundary condition u ( x ) = 0 for one arbitrary chosen boundary edge which is not an electrode to overcome non-uniqueness. ◮ Rotate electrodes where current flows and repeatedly solve the PDE until the starting position is reached. TUC · 2nd November 2017 · Christopher Hofmann 12 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Simplified Model Recalling the model geometry Figure: Specimen with electrodes and finite element grid. TUC · 2nd November 2017 · Christopher Hofmann 13 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Simplified Model Discretized Forward Operator Full set of measurements, ◮ Ω discretized into n ‘stripes’. ◮ Electrodes are rotated. ◮ The associated elliptic problem is solved in a repeated manner. This results in the following nonlinear operator: σ = ( σ 1 , ..., σ n ) T ∈ R n F ( σ ) ∈ R 8 × 8 �→ Note: Every column of F is one set of measurements. TUC · 2nd November 2017 · Christopher Hofmann 14 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Simplified Model Least-Squares Minimization Assume that σ ∗ ∈ R n + is the ‘true conductivity vector’ to be identified: σ δ � F ( σ ) − F δ ( σ ∗ ) � F LS = arg min ( ∗ ) σ ∈ Q where ◮ Q ⊂ R n + is the set of admissible solutions. ◮ � · � F designates the Frobenius norm. ◮ F δ ( σ ∗ ) indicates noisy data associated with some noise level δ > 0 . TUC · 2nd November 2017 · Christopher Hofmann 15 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Simplified Model Noise Model Additive noise model: F δ ( σ ∗ ) = F ( σ ∗ ) + E with E = ( ε ij ) ∈ R 8 × 8 , ǫ ij i.i.d. and ǫ ij ∼ N (0 , d 2 ) Chose d so that: � � F δ ( σ ∗ ) − F ( σ ∗ ) � 2 � F = δ 2 , E � F ( σ ∗ ) � 2 F which leads to: d = δ 8 � F ( σ ∗ ) � F . TUC · 2nd November 2017 · Christopher Hofmann 16 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Numerical Examples Two unknown conductivities Ω discretized with a stripe structure, here n=2, with conductivities σ 1 and σ 2 . Forward calculations for all admissible values in Q = { ( σ 1 , σ 2 ) ∈ [10 , 75] × [5 , 46] } . Figure: Material ‘stripes’ with two unknown conductivities. TUC · 2nd November 2017 · Christopher Hofmann 17 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Numerical Examples Two unknown conductivities Discrepancy norm � F ( σ ) − F ( σ ∗ ) � F depending on σ = ( σ 1 , σ 2 ) T for σ ∗ = (37 . 7 , 7 . 9) T . Figure: Perspective drawing and level sets of discrepancy norm � F ( σ ) − F ( σ ∗ ) � F depending on σ = ( σ 1 , σ 2 ) T . TUC · 2nd November 2017 · Christopher Hofmann 18 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Numerical Examples Two unknown conductivities Discrepancy norm � F ( σ ) − F ( σ ∗ ) � F depending on σ = ( σ 1 , σ 2 ) T for σ ∗ = (6 , 5) T . Figure: Level sets of discrepancy norm � F ( σ ) − F ( σ ∗ ) � F depending on σ = ( σ 1 , σ 2 ) T . TUC · 2nd November 2017 · Christopher Hofmann 19 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Numerical Examples Five unknown conductivities Ω discretized in n=5 ’stripes’. Numerical evidence suggests that very different conductivies lead to very similar measurements: Figure: left: σ (1) = (4 . 26 , 17 . 33 , 7 . 65 , 0 . 99 , 1 . 00) T , right: σ (2) = (4 . 27 , 23 . 87 , 4 . 34 , 50 . 00 , 28 . 99) T , � F ( σ (1) ) − F ( σ (2) ) � 2 F = 0 . 000099 . TUC · 2nd November 2017 · Christopher Hofmann 20 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
Numerical Examples Five unknown conductivities f ( λ ) := � F ( σ ∗ + λ ( σ (3) − σ ∗ )) − F δ ( σ ∗ ) � F , λ ∈ [ − 5 , 1] , for σ ∗ = (7 . 53 , 22 . 23 , 14 . 28 , 4 . 26 , 4 . 99) T and σ (3) = (7 . 53 , 45 . 09 , 12 . 63 , 4 . 26 , 4 . 99) T . Figure: Graph of f ( λ ) for λ ∈ [ − 5 , 1] without noise ( δ = 0 ) and with 5% noise ( δ = 0 . 05 ). TUC · 2nd November 2017 · Christopher Hofmann 21 / 28 www.tu-chemnitz.de/mathematik/inverse probleme/
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